Sfera, tekislik va giperbolik tekislikdagi bir tekis qiya ro'yxatlar - Lists of uniform tilings on the sphere, plane, and hyperbolic plane - Wikipedia
Yilda geometriya, sfera, evklid tekisligi va giperbolik tekislikda ko'plab bir xil tekisliklarni bajarish mumkin Wythoff qurilishi tri / p, π / q va π / r kabi ichki burchaklar bilan aniqlangan (p q r) asosiy uchburchak ichida. Maxsus holatlar - bu to'rtburchaklar (p q 2). Yagona echimlar bitta generator punkti tomonidan hayoliy uchburchak ichida 7 ta o'ringa, 3 burchakka, 3 qirraga va uchburchakning ichki qismiga o'rnatiladi. Barcha tepaliklar generatorda yoki uning aks etgan nusxasida mavjud. Chegaralar generator nuqtasi va uning oynadagi tasviri o'rtasida mavjud. Asosiy uchburchakning markazida joylashgan 3 tagacha yuz turi mavjud. To'g'ri uchburchak domenlari 1 ta yuz turiga ega bo'lishi mumkin, ular odatiy shakllarni yaratadilar, umumiy uchburchaklar esa kamida 2 ta uchburchak turlariga ega bo'lib, eng yaxshisi kvazirgular plitkalarga olib keladi.
Ushbu yagona echimlarni ifodalash uchun turli xil belgilar mavjud, Wythoff belgisi, Kokseter diagrammasi, va Kokseterning t-notasi.
Oddiy plitkalar tomonidan ishlab chiqarilgan Mobius uchburchagi p, q, r, butun sonlari bilan Shvarts uchburchagi p, q, r ratsional sonlariga ruxsat bering va ruxsat bering yulduz ko'pburchagi yuzlari va bir-birining ustiga chiqadigan elementlari bor.
7 generator punkti
Har bir to'plam bilan yettita generator (va bir nechta maxsus shakllar):
Umumiy | To'g'ri uchburchak (r = 2) | |||||||
---|---|---|---|---|---|---|---|---|
Tavsif | Wythoff belgi | Tepalik konfiguratsiya | Kokseter diagramma ![]() | Wythoff belgi | Tepalik konfiguratsiya | Schläfli belgi | Kokseter diagramma ![]() ![]() ![]() ![]() ![]() | |
muntazam va quasiregular | q | p r | (p.r)q | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | q | p 2 | pq | {p, q} | ![]() ![]() ![]() ![]() ![]() | |
p | q r | (q.r)p | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | p | q 2 | qp | {q, p} | ![]() ![]() ![]() ![]() ![]() | ||
r | p q | (q.p)r | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | 2 | p q | (q.p)² | r {p, q} | t1{p, q} | ![]() ![]() ![]() ![]() ![]() | |
kesilgan va kengaytirilgan | q r | p | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | q 2 | p | t {p, q} | t0,1{p, q} | ![]() ![]() ![]() ![]() ![]() | ||
p r | q | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | p 2 | q | p. 2q.2q | t {q, p} | t0,1{q, p} | ![]() ![]() ![]() ![]() ![]() | ||
p q | r | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | p q | 2 | rr {p, q} | t0,2{p, q} | ![]() ![]() ![]() ![]() ![]() | |||
tekis yuzli | p q r | | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | p q 2 | | tr {p, q} | t0,1,2{p, q} | ![]() ![]() ![]() ![]() ![]() | ||
p q (r s) | | - | p 2 (r s) | | 2p.4.-2p.4/3 | - | ||||
qotib qolish | | p q r | ![]() ![]() ![]() ![]() ![]() ![]() ![]() | | p q 2 | sr {p, q} | ![]() ![]() ![]() ![]() ![]() | |||
| p q r s | - | - | - | - |
Uchta maxsus holat mavjud:
- - Bu aralashmasi va , ikkalasi ham birgalikda foydalanadigan yuzlarni o'z ichiga oladi.
- - Snub shakllari (o'zgaruvchan) ushbu boshqa ishlatilmaydigan belgi bilan beriladi.
- - uchun noyob snub shakli U75 bu Wythoff tomonidan tuzilmaydi.
Simmetriya uchburchagi
Ning aks ettirishning 4 ta simmetriya klassi mavjud soha, va uchta Evklid samolyoti. Ulardan bir nechtasi cheksiz ko'p bunday naqshlar giperbolik tekislik shuningdek ro'yxatga olingan. (Giperbolik yoki Evklid plitkalarini belgilaydigan sonlarning birortasini ko'paytirish yana bir giperbolik qoplamani hosil qiladi.)
Nuqta guruhlari:
- (p 2 2) dihedral simmetriya, (buyurtma )
- (3 3 2) tetraedral simmetriya (buyurtma 24)
- (4 3 2) oktahedral simmetriya (buyurtma 48)
- (5 3 2) ikosahedral simmetriya (buyurtma 120)
Evklid (afin) guruhlari:
- (4 4 2) * 442 simmetriya: 45 ° -45 ° -90 ° uchburchak
- (6 3 2) *632 simmetriya: 30 ° -60 ° -90 ° uchburchak
- (3 3 3) *333 simmetriya: 60 ° -60 ° -60 ° uchburchak
Giperbolik guruhlar:
- (7 3 2) *732 simmetriya
- (8 3 2) *832 simmetriya
- (4 3 3) *433 simmetriya
- (4 4 3) *443 simmetriya
- (4 4 4) *444 simmetriya
- (5 4 2) *542 simmetriya
- (6 4 2) *642 simmetriya
- ...
Ikki tomonlama sferik | Sharsimon | ||||||
---|---|---|---|---|---|---|---|
D.2 soat | D.3 soat | D.4 soat | D.5 soat | D.6 soat | Td | Oh | Menh |
*222 | *322 | *422 | *522 | *622 | *332 | *432 | *532 |
![]() (2 2 2) | ![]() (3 2 2) | ![]() (4 2 2) | ![]() (5 2 2) | ![]() (6 2 2) | ![]() (3 3 2) | ![]() (4 3 2) | ![]() (5 3 2) |
Yuqoridagi simmetriya guruhlariga faqat sferadagi butun sonli echimlar kiradi. Shvarts uchburchaklarining ro'yxati ratsional sonlarni o'z ichiga oladi va ularning echimlarining to'liq to'plamini aniqlaydi konveks bo'lmagan bir xil polyhedra.
p4m | p3m | p6m |
---|---|---|
*442 | *333 | *632 |
![]() (4 4 2) | ![]() (3 3 3) | ![]() (6 3 2) |
*732 | *542 | *433 |
---|---|---|
![]() (7 3 2) | ![]() (5 4 2) | ![]() (4 3 3) |
Yuqoridagi plitkalarda har bir uchburchak juft va g'alati akslantirishlar bilan bo'yalgan asosiy domen hisoblanadi.
Xulosa sharsimon, evklid va giperbolik plitkalar
Wythoff konstruktsiyasi tomonidan yaratilgan tanlangan plitkalar quyida keltirilgan.
Sferik plitkalar (r = 2)
(p q 2) | Ota-ona | Qisqartirilgan | Tuzatilgan | Bitruncated | Birlashtirilgan (dual) | Kantellatsiya qilingan | Hamma narsa (Kantritratsiya qilingan) | Snub |
---|---|---|---|---|---|---|---|---|
Wythoff belgi | q | 2-bet | 2 q | p | 2 | p q | 2 p | q | p | q 2 | p q | 2018-04-02 121 2 | p q 2 | | | p q 2 |
Schläfli belgi | ||||||||
{p, q} | t {p, q} | r {p, q} | t {q, p} | {q, p} | rr {p, q} | tr {p, q} | sr {p, q} | |
t0{p, q} | t0,1{p, q} | t1{p, q} | t1,2{p, q} | t2{p, q} | t0,2{p, q} | t0,1,2{p, q} | ||
Kokseter diagramma | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() |
Tepalik shakli | pq | q.2p.2p | (p.q)2 | p. 2q.2q | qp | p. 4.q.4 | 4.2p.2q | 3.3.p. 3.q |
![]() (3 3 2) | ![]() {3,3} | ![]() (3.6.6) | ![]() (3.3a.3.3a) | ![]() (3.6.6) | ![]() {3,3} | ![]() (3a.4.3b.4) | ![]() (4.6a.6b) | ![]() (3.3.3a.3.3b) |
![]() (4 3 2) | ![]() {4,3} | ![]() (3.8.8) | ![]() (3.4.3.4) | ![]() (4.6.6) | ![]() {3,4} | ![]() (3.4.4a.4) | ![]() (4.6.8) | ![]() (3.3.3a.3.4) |
![]() (5 3 2) | ![]() {5,3} | ![]() (3.10.10) | ![]() (3.5.3.5) | ![]() (5.6.6) | ![]() {3,5} | ![]() (3.4.5.4) | ![]() (4.6.10) | ![]() (3.3.3a.3.5) |
Ba'zi bir-biriga o'xshash sharsimon plitkalar (r = 2)
- To'liq ro'yxat, shu jumladan holatlar uchun r ≠ 2, qarang Shvarts uchburchagi bir xil ko'p qirrali ro'yxati.
Plitkalar ko'rsatilgan polyhedra. Ba'zi shakllar buzilib ketgan, ular uchun qavslar bilan berilgan tepalik raqamlari, bir-birining ustiga chiqib ketgan qirralar yoki tepaliklar bilan.
(p q 2) | Jamg'arma. uchburchak | Ota-ona | Qisqartirilgan | Tuzatilgan | Bitruncated | Birlashtirilgan (dual) | Kantellatsiya qilingan | Hamma narsa (Kantritratsiya qilingan) | Snub |
---|---|---|---|---|---|---|---|---|---|
Wythoff belgisi | q | 2-bet | 2 q | p | 2 | p q | 2 p | q | p | q 2 | p q | 2018-04-02 121 2 | p q 2 | | | p q 2 | |
Schläfli belgisi | |||||||||
{p, q} | t {p, q} | r {p, q} | t {q, p} | {q, p} | rr {p, q} | tr {p, q} | sr {p, q} | ||
t0{p, q} | t0,1{p, q} | t1{p, q} | t1,2{p, q} | t2{p, q} | t0,2{p, q} | t0,1,2{p, q} | |||
Kokseter - Dinkin diagrammasi | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | |
Tepalik shakli | pq | (q.2p.2p) | (p.q.p.q) | (2q.2q bet) | qp | (4.q.4-bet). | (4.2p.2q) | (3.3p. 3.q) | |
Ikosahedral (5/2 3 2) | ![]() {3,5/2} | ![]() (5/2.6.6) | ![]() (3.5/2)2 | ![]() [3.10/2.10/2] | ![]() {5/2,3} | ![]() [3.4.5/2.4] | ![]() [4.10/2.6] | ![]() (3.3.3.3.5/2) | |
Ikosahedral (5 5/2 2) | ![]() {5,5/2} | ![]() (5/2.10.10) | ![]() (5/2.5)2 | ![]() [5.10/2.10/2] | ![]() {5/2,5} | ![]() (5/2.4.5.4) | ![]() [4.10/2.10] | ![]() (3.3.5/2.3.5) |
Dihedral simmetriya (q = r = 2)
Bilan sferik plitkalar dihedral simmetriya hamma uchun mavjud ko'plari bilan digon degenerativ polyhedraga aylanadigan yuzlar. Sakkiz shakldan ikkitasi (Rektifikatsiya qilingan va kantillangan) replikatsiya bo'lib, jadvalda o'tkazib yuborilgan.
(p 2 2) Asosiy domen | Ota-ona | Qisqartirilgan | Bitruncated (qisqartirilgan dual) | Birlashtirilgan (dual) | Hamma narsa (Kantritratsiya qilingan) | Snub | |||
---|---|---|---|---|---|---|---|---|---|
Kengaytirilgan Schläfli belgisi | |||||||||
{p, 2} | t {p, 2} | t {2, p} | {2, p} | tr {p, 2} | sr {p, 2} | ||||
t0{p, 2} | t0,1{p, 2} | t1,2{p, 2} | t2{p, 2} | t0,1,2{p, 2} | |||||
Wythoff belgisi | 2 | 2-bet | 2 2 | p | 2 p | 2018-04-02 121 2 | p | 2018-04-02 2 121 2 | p 2 2 | | | p 2 2 | |||
Kokseter - Dinkin diagrammasi | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | |||
Tepalik shakli | p² | (2.2p.2p) | (4.4.p) | 2p | (4.2p.4) | (3.3-bet 3) | |||
![]() (2 2 2) V2.2.2 | ![]() {2,2} | ![]() 2.4.4 | 4.4.2 | ![]() {2,2} | ![]() 4.4.4 | ![]() 3.3.3.2 | |||
![]() (3 2 2) V3.2.2 | ![]() {3,2} | ![]() 2.6.6 | ![]() 4.4.3 | ![]() {2,3} | ![]() 4.4.6 | ![]() 3.3.3.3 | |||
![]() (4 2 2) V4.2.2 | ![]() {4,2} | 2.8.8 | ![]() 4.4.4 | ![]() {2,4} | ![]() 4.4.8 | ![]() 3.3.3.4 | |||
![]() (5 2 2) V5.2.2 | ![]() {5,2} | 2.10.10 | ![]() 4.4.5 | ![]() {2,5} | ![]() 4.4.10 | ![]() 3.3.3.5 | |||
![]() (6 2 2) V6.2.2 | ![]() {6,2} | ![]() 2.12.12 | ![]() 4.4.6 | ![]() {2,6} | ![]() 4.4.12 | ![]() 3.3.3.6 | |||
... |
Evklid va giperbolik plitkalar (r = 2)
Ba'zi bir giperbolik plitkalar berilgan va a shaklida ko'rsatilgan Poincaré disk proektsiya.
Evklid va giperbolik plitkalar (r > 2)
The Kokseter - Dinkin diagrammasi chiziqli shaklda berilgan, garchi u aslida uchburchak bo'lsa ham, oxirgi segment r birinchi tugunga ulanadi.
Shuningdek qarang
- Muntazam politop
- Muntazam ko'pburchak
- Bir xil plitkalar ro'yxati
- Giperbolik tekislikdagi bir tekis plitkalar
- Bir xil polyhedra ro'yxati
- Shvarts uchburchagi bir xil ko'p qirrali ro'yxati
Adabiyotlar
- Kokseter Muntazam Polytopes, Uchinchi nashr, (1973), Dover nashri, ISBN 0-486-61480-8 (V bob: Kaleydoskop, Bo'lim: 5.7 Vaytof qurilishi)
- Kokseter Geometriyaning go'zalligi: o'n ikkita esse, Dover Publications, 1999, ISBN 0-486-40919-8 (3-bob: Uythoffning yagona politoplar uchun qurilishi)
- Kokseter, Longuet-Xiggins, Miller, Yagona polyhedra, Fil. Trans. 1954, 246 A, 401-50.
- Venninger, Magnus (1974). Polyhedron modellari. Kembrij universiteti matbuoti. ISBN 0-521-09859-9. 9-10 betlar.
Tashqi havolalar
- Vayshteyn, Erik V. "Wythoff belgisi". MathWorld.
- Wythoff belgisi
- Wythoff belgisi[doimiy o'lik havola ]
- Wythoff qurilish uslubidan foydalangan holda bir xil polidralarni namoyish qilish uchun Greg Eganning appleti
- Vaythoffning qurilish usulini Shadertoy taqdim etadi
- KaleidoTile 3 Tomonidan Windows uchun bepul o'quv dasturi Jeffri Uiks sahifadagi ko'plab rasmlarni yaratgan.
- Xetch, Don. "Giperbolik planar tessellations".
![]() | Ushbu maqola matematikaga oid narsalarni o'z ichiga oladi ro'yxatlar ro'yxati. |